1.9 Electron Band Calculations in Three Dimensions
1.9.3 The Nearly Free Electron Approximation
Attheoppositeextremefromthe tight-bindingapproximation,onecanmake thenearly freeapproximationforelectronsinbandsthat arisefromatomicorbitalswithverylarge overlap. In this case, theelectron states are almost the sameas free plane waves. This approximationcanworkforthenearlyfreeelectronsinupperstateseveniftherearetightly boundelectronsinlower,corestates.Theeffectofthecoreelectronsistakenintoaccount justasachangeofthetotalclassicalchargeofthenucleus.
Inthisapproximation,webeginwiththeversionofBloch’stheoremgivenin(1.6.4), ψn´k(´r) = 1
√V
ằ
´Q
Cn(´k − ´Q)ei(´k−´Q)ã´r, (1.9.21)
where the ´Qs are reciprocal lattice vectors. The nearly free electron approximation amountsto assumingthat thewave functionis nearly equaltoa planewave, ei´kã´r.This meansthattheleadingtermintheexpansionis Cn(´k) ≈ 1,whichcorrespondsto ´Q = 0, andhigher-ordertermsofthisexpansionaresmallcomparedtothisterm.
Wealsowritetheperiodicpotential U(´r)asaFourierseries, U(´r) =ằ
´Q
U( ´Q)e−i ´Qã´r. (1.9.22)
Wewillassumethat U(´Q = 0)iszero,sincethiscorrespondstoaconstanttermwhichwe canalwaysremovebychangingthedefinitionofzeropotentialenergy.Substitutingthese definitionsintotheSchrửdingerequation,wehave
53 1.9 ElectronBandCalculationsinThreeDimensions
¿
− ±2
2m∇2+U(´r) − En(´k)À
ψn´k(´r) = 0
= √1 V
ằ
´Q
⎡
⎣ ±2
2m|´k − ´Q|2+ằ
´Qá
U(´Qá)e−i ´Qáã´r− En(´k)
⎤
⎦ Cn(´k − ´Q)ei(´k− ´Q)ã´r. (1.9.23) Multiplyingby(1/V)e−i(´k−´Qáá)ã´randintegratingover allspaceallowsustoeliminatethe exponentialfactorsandoneofthesummations,becauseweknowthat(1/V)Á
d3rei´kã´r= δ´k,0(seeAppendixC).Weobtain
¿±2
2m|´k − ´Q|2− En(´k) Â
Cn(´k − ´Q) +ằ
´Qá
U(´Q − ´Qá)Cn(´k − ´Qá) = 0. (1.9.24) Thisisthemainequationforthenearlyfreeelectronapproximation.Itisstillexact,butas withthetight-bindingmodel,wecansolveitapproximatelybytruncatingthesumover ´Qá toonlynearneighborsof ´Q = 0,ontheassumptionthat Cn(´k − ´Qá)fallsoffrapidlyfor increasing ´Qá.Inthiscase,wetruncateinreciprocalspaceratherthanrealspace,because awavefunctionspreadoutinrealspace,likeaplanewave,islocalizedin k-space.
Wecangetafeelforthismodelbytakingjustonenearestneighbor, ´Qá= ´Q0.Ifweset
´Q = 0,wehave
¿±2k2
2m − En(´k) Â
Cn(´k) + U(− ´Qá)Cn(´k − ´Qá) = 0. (1.9.25) Wecanalsoset ´Q = ´Q0in(1.9.24),whichgivesus,tofirstorder,
ñ2|´k − ´Qá|2
2m − En(´k) Ä
Cn(´k − ´Qá) + U(´Qá)Cn(´k) = 0. (1.9.26) Theseequationscanbewrittenas
⎛
⎜⎜
⎝
±2k2
2m U∗(´Q0) U(´Q0) ±2|´k − ´Q0|2
2m
⎞
⎟⎟
⎠
² Cn(´k) Cn(´k − ´Q0)
³
= En(´k)
² Cn(´k) Cn(´k − ´Q0)
³ ,
(1.9.27) where we have used the relation U(− ´Q0) = U∗( ´Q0), which follows from the proper- tiesoftheinverseFouriertransformwhen U(´r)isreal. Wethushaveatwo-dimensional eigenvalueproblem.Setting ´k = ´Q0/2 − ±´k,wefindtheeigenvalues
E(´k) = ±2|±´k|2
2m +±2(Q0/2)2
2m ±
´²±2|´Q0ã ±´k|
2m
³2
+|U(´Q0)|2. (1.9.28) Inotherwords,in theregion neara zone boundary,anenergygap opens,just as inthe Kronig–Penneymodel.
wave vector coordinate
Energy
T K T’ M
T MT’K Γ
Γ Γ
Σ
Σ
±Fig.1.30 Solidlines:Energybandsofahexagonallatticeinthenearlyfreeelectronapproximation,foundbydiagonalizinga 19 ì 19matrixfor Cn(´k)andthe18nearestneighbors Cn(´k − ´Qá)inthetwo-dimensionalhexagonalreciprocal lattice.Dashedlines:freeelectronenergy E = ±2k2/2m inthereducedzoneofthesamehexagonallattice.
Figure1.30 shows anumerical solutionof theelectronbands for ahexagonalcrystal inthenearlyfreeelectronapproximation.As seeninthisfigure,theenergyofthebands tracksthefree-particleenergy E(´k) = ±2k2/2m.
Thismodelhelpsustoseewhatwouldhappentothebandsifweusedthewrongunit cell.InSection1.4,wesawthatifwedecidedtoviewalatticewithcellsize a asalattice withcellsize2a andatwo-atombasis,thestructurefactorwouldensurethatnoextrapeaks wouldbepredicted.Itshouldalsobethecasethatifwechoseadouble-sizeunitcellfor computingthebandstructure,weshouldstillgetthesameanswer.
In(1.9.28),weseethatthegapenergydependsontheFouriercomponentofthepotential thatcorrespondstothereciprocallatticevector ´Q0.Ifwechosethewronglatticespacing, 2a, anda two-atombasis,then thiswouldimply azone boundaryat Q = π/2a,which mightleadustoexpectagapthere.ThiswouldonlyoccurifthereisaFouriercomponent U(´Q/2),however.But theperiodicfunctionsonlyhaveFouriercomponents correspond- ingtothereciprocallatticevectors.Therefore,althoughwecoulddrawtheenergybands in areducedzone withboundary ´Q/2, therewould benogapin thebands at thatzone boundary.
Higher-order Brillouin zones.Aswehaveseen,inthenearlyfreeelectronapproxima- tion,theelectronbandsarenearlyequaltotheenergyofafreeelectroninvacuum.When weplot thisfreeelectron energyinthereduced-zonescheme,we getaseriesofhigher- energy bands.In three dimensions,the shape ofthese higher-energybands canbevery complicated.
55 1.9 ElectronBandCalculationsinThreeDimensions
Aconstructionintermsofadditional,higher-orderBrillouinzonescanhelpustovisu- alizewhatthesehigher-energybandswilllooklike.Inessence,thehigher-orderBrillouin zonepictureisjustbasedonthefactthatanypartofthereciprocallatticecanbemapped backintothefirstBrillouinzoneinthereduced-zonescheme,bysubtractingoraddingan integernumberofreciprocallatticevectors.Thehigher-orderBrillouinzonestelluswhich partsofreciprocalspacearemappedtowhichpartsofthefirstBrillouinzone.
Therecipefordrawingthehigher-orderBrillouinzonesisasfollows:
• Drawthereciprocallattice.
• DrawalltheBraggplanes,whichconsistofallplaneswhichbisectalinebetweenthe originandanyotherreciprocallatticepoint.
• ThefirstBrillouinzoneisthesetofallpointsthatcanbereachedfromtheoriginwithout passingthroughanyBraggplanes.ThesecondBrillouinzoneisthesetofallpointsthat canbereachedbypassingthroughonlyoneBraggplane,thethirdBrillouinzoneisthe setthatcanbereachedbypassingthroughtwoBraggplanes,etc.
• Thepartsofahigher-orderzonethatlieoutsidethefirstzonecanbemappedbackinto thefirstBrillouinzoneinthereduced-zonescheme.
It can beproven that mapping each part of a higher-order Brillouin zone to the first Brillouinzoneinthereduced-zoneschemewillalwayscompletelyfillupthefirstBrillouin zone;inotherwords,eachhigher-orderzoneisalsoavalidprimitivecellforthereciprocal lattice.
Inthecaseofnearlyfreeelectrons,theenergybandsarejustaperturbationofthefree- electronenergysurface.Therefore,thehigher-energybandswillbeacloseapproximation ofthe free-electron energyin the reduced-zonescheme. Figure 1.31shows an example ofhowthe high-orderBrillouin zonesgeneratethehigh-energybands inthenearlyfree electronapproximation,foratwo-dimensionalhexagonallattice.Asseeninthisfigure,the higher-energybandshave folds whicharise from theshape ofthe original free-electron energyband.
Exercise1.9.5 ConstructthefirstfourBrillouinzonesforatwo-dimensionalsquarelattice, andshowthatmappingeachpartofthehigher-orderzoneintothefirstBrillouinzone fillsuptheentirefirstzone.